Scattering and inverse scattering in the presence of complex background media

Monk, Peter B.Scattering and inverse scattering theory plays a central role in mathematical physics. For example, through the use of acoustic or electromagnetic waves, one can detect and identify objects that are hard to or cannot directly be observed as well as obtain information about the material properties of objects of interest. However, in practical applications, the presence of complex background media in which the problems are considered restricts us from applying directly the existing theoretical results and numerical methods. This constraint requires delicate modification of well-established theory and development of alternate computational approaches. In this thesis, we investigate the applicability of qualitative methods in inverse scattering theory for obtaining material properties and recovering shapes of unknown objects by using time-harmonic electromagnetic waves under different geometrical configurations. In particular, we first consider a 2D model where a bounded dielectric scatterer sits on an infinite metallic substrate. This is a model problem for non-destructive testing of aircraft coatings. We validate the application of the Linear Sampling Method (LSM) for detecting special frequencies called transmission eigenvalues for both isotropic and anisotropic media. Then we move to a 3D model where a bounded perfectly electric conducting object is located inside an infinite long perfectly electric conducting waveguide and justify the application of the LSM for reconstructing the shape of the object. For both cases, we show that additional work needs to be done in order to recast standard results from scattering and inverse scattering theory to the model problems we consider. Also, this work gives an idea of the effort needed to adapt academic research to industrial applications.University of Delaware, Department of Mathematical SciencesPh.D